多轨迹相互作用粒子随机系统中相互作用核的学习
Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories
摘要：
我们考虑相互作用的粒子或代理的随机系统，与动力学的相互作用内核，只依赖于成对的距离。我们研究的问题推断这个相互作用内核的粒子的位置，在连续或离散的时间，沿多个独立的轨迹的观测。我们引入了一个非参数推理方法，这个逆问题的基础上，约束到合适的假设空间自适应数据的正则化最大似然估计。我们表明，一个adjuvity条件，使我们能够控制这个问题的条件数，并证明我们的估计的一致性，事实上，它收敛在一个接近最优的学习率，等于最小-最大率的$1$维非参数回归。特别地，该速率与状态空间的维度无关，该维度通常非常高。我们还分析了离散时间观测的情况下的离散化误差，表明它是$1/2$阶的观测之间的时间间隔。这个术语，当大时，主导采样误差和近似误差，防止估计的收敛。最后，我们展示了一个有效的并行算法来构建估计数据，我们证明了我们的算法的有效性与原型系统，包括随机意见动态和Lennard-Jones模型的数值测试。
英文摘要：
We consider stochastic systems of interacting particles or agents, with dynamics determined by an interaction kernel which only depends on pairwise distances. We study the problem of inferring this interaction kernel from observations of the positions of the particles, in either continuous or discrete time, along multiple independent trajectories. We introduce a nonparametric inference approach to this inverse problem, based on a regularized maximum likelihood estimator constrained to suitable hypothesis spaces adaptive to data. We show that a coercivity condition enables us to control the condition number of this problem and prove the consistency of our estimator, and that in fact it converges at a near-optimal learning rate, equal to the min-max rate of $1$-dimensional non-parametric regression. In particular, this rate is independent of the dimension of the state space, which is typically very high. We also analyze the discretization errors in the case of discrete-time observations, showing that it is of order $1/2$ in terms of the time gaps between observations. This term, when large, dominates the sampling error and the approximation error, preventing convergence of the estimator. Finally, we exhibit an efficient parallel algorithm to construct the estimator from data, and we demonstrate the effectiveness of our algorithm with numerical tests on prototype systems including stochastic opinion dynamics and a Lennard-Jones model.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
